Stabilization for vibrating plate with singular structural damping

We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.